math.sqrt on complex, imaginary part, im greater than 0 branch

Percentage Accurate: 41.7% → 88.7%

Time: 5.1s

Alternatives: 6

Speedup: 2.0×

Specification

\[im > 0\]

Math

\[\begin{array}{l} \\ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* 0.5 (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re)))))

C

double code(double re, double im) {
	return 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) - re)));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.5d0 * sqrt((2.0d0 * (sqrt(((re * re) + (im * im))) - re)))
end function

Java

public static double code(double re, double im) {
	return 0.5 * Math.sqrt((2.0 * (Math.sqrt(((re * re) + (im * im))) - re)));
}

Python

def code(re, im):
	return 0.5 * math.sqrt((2.0 * (math.sqrt(((re * re) + (im * im))) - re)))

Julia

function code(re, im)
	return Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(Float64(Float64(re * re) + Float64(im * im))) - re))))
end

MATLAB

function tmp = code(re, im)
	tmp = 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) - re)));
end

Wolfram

code[re_, im_] := N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 41.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* 0.5 (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re)))))

C

double code(double re, double im) {
	return 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) - re)));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.5d0 * sqrt((2.0d0 * (sqrt(((re * re) + (im * im))) - re)))
end function

Java

public static double code(double re, double im) {
	return 0.5 * Math.sqrt((2.0 * (Math.sqrt(((re * re) + (im * im))) - re)));
}

Python

def code(re, im):
	return 0.5 * math.sqrt((2.0 * (math.sqrt(((re * re) + (im * im))) - re)))

Julia

function code(re, im)
	return Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(Float64(Float64(re * re) + Float64(im * im))) - re))))
end

MATLAB

function tmp = code(re, im)
	tmp = 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) - re)));
end

Wolfram

code[re_, im_] := N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 88.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 170000:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 170000.0)
   (* 0.5 (sqrt (* 2.0 (- (hypot re im) re))))
   (* 0.5 (* im (pow re -0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 170000.0) {
		tmp = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	} else {
		tmp = 0.5 * (im * pow(re, -0.5));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 170000.0) {
		tmp = 0.5 * Math.sqrt((2.0 * (Math.hypot(re, im) - re)));
	} else {
		tmp = 0.5 * (im * Math.pow(re, -0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 170000.0:
		tmp = 0.5 * math.sqrt((2.0 * (math.hypot(re, im) - re)))
	else:
		tmp = 0.5 * (im * math.pow(re, -0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 170000.0)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(hypot(re, im) - re))));
	else
		tmp = Float64(0.5 * Float64(im * (re ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 170000.0)
		tmp = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	else
		tmp = 0.5 * (im * (re ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 170000.0], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * N[Power[re, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.7e5

   1. Initial program 53.0%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   2. Step-by-step derivation

      1. hypot-def 93.1%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\mathsf{hypot}\left(re, im\right)} - re\right)} \]

   3. Simplified 93.1%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}} \]

   if 1.7e5 < re

   1. Initial program 10.4%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   2. Step-by-step derivation

      1. hypot-def 36.7%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\mathsf{hypot}\left(re, im\right)} - re\right)} \]

   3. Simplified 36.7%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}} \]

   4. Taylor expanded in re around inf 58.7%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{{im}^{2}}{re}}} \]
   5. Step-by-step derivation

      1. unpow2 58.7%

         \[\leadsto 0.5 \cdot \sqrt{\frac{\color{blue}{im \cdot im}}{re}} \]

   6. Simplified 58.7%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{im \cdot im}{re}}} \]
   7. Step-by-step derivation

      1. div-inv 58.7%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\left(im \cdot im\right) \cdot \frac{1}{re}}} \]

      2. sqrt-prod 71.5%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{im \cdot im} \cdot \sqrt{\frac{1}{re}}\right)} \]

      3. sqrt-prod 83.7%

         \[\leadsto 0.5 \cdot \left(\color{blue}{\left(\sqrt{im} \cdot \sqrt{im}\right)} \cdot \sqrt{\frac{1}{re}}\right) \]

      4. add-sqr-sqrt 83.9%

         \[\leadsto 0.5 \cdot \left(\color{blue}{im} \cdot \sqrt{\frac{1}{re}}\right) \]

      5. inv-pow 83.9%

         \[\leadsto 0.5 \cdot \left(im \cdot \sqrt{\color{blue}{{re}^{-1}}}\right) \]

      6. sqrt-pow1 84.0%

         \[\leadsto 0.5 \cdot \left(im \cdot \color{blue}{{re}^{\left(\frac{-1}{2}\right)}}\right) \]

      7. metadata-eval 84.0%

         \[\leadsto 0.5 \cdot \left(im \cdot {re}^{\color{blue}{-0.5}}\right) \]

   8. Applied egg-rr 84.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot {re}^{-0.5}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 170000:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \]

Alternative 2: 77.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -6.4 \cdot 10^{+47}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{-9}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -6.4e+47)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 1.9e-9)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (* 0.5 (* im (pow re -0.5))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -6.4e+47) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 1.9e-9) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else {
		tmp = 0.5 * (im * pow(re, -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-6.4d+47)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 1.9d-9) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else
        tmp = 0.5d0 * (im * (re ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -6.4e+47) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 1.9e-9) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else {
		tmp = 0.5 * (im * Math.pow(re, -0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -6.4e+47:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 1.9e-9:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	else:
		tmp = 0.5 * (im * math.pow(re, -0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -6.4e+47)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 1.9e-9)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	else
		tmp = Float64(0.5 * Float64(im * (re ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -6.4e+47)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 1.9e-9)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	else
		tmp = 0.5 * (im * (re ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -6.4e+47], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.9e-9], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * N[Power[re, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -6.4e47

   1. Initial program 23.4%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   2. Step-by-step derivation

      1. hypot-def 100.0%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\mathsf{hypot}\left(re, im\right)} - re\right)} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}} \]

   4. Taylor expanded in re around -inf 81.7%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-4 \cdot re}} \]
   5. Step-by-step derivation

      1. *-commutative 81.7%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   6. Simplified 81.7%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   if -6.4e47 < re < 1.90000000000000006e-9

   1. Initial program 65.1%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]

   2. Taylor expanded in re around 0 77.1%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{im} - re\right)} \]

   if 1.90000000000000006e-9 < re

   1. Initial program 10.1%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   2. Step-by-step derivation

      1. hypot-def 37.6%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\mathsf{hypot}\left(re, im\right)} - re\right)} \]

   3. Simplified 37.6%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}} \]

   4. Taylor expanded in re around inf 57.0%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{{im}^{2}}{re}}} \]
   5. Step-by-step derivation

      1. unpow2 57.0%

         \[\leadsto 0.5 \cdot \sqrt{\frac{\color{blue}{im \cdot im}}{re}} \]

   6. Simplified 57.0%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{im \cdot im}{re}}} \]
   7. Step-by-step derivation

      1. div-inv 57.0%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\left(im \cdot im\right) \cdot \frac{1}{re}}} \]

      2. sqrt-prod 69.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{im \cdot im} \cdot \sqrt{\frac{1}{re}}\right)} \]

      3. sqrt-prod 81.8%

         \[\leadsto 0.5 \cdot \left(\color{blue}{\left(\sqrt{im} \cdot \sqrt{im}\right)} \cdot \sqrt{\frac{1}{re}}\right) \]

      4. add-sqr-sqrt 82.1%

         \[\leadsto 0.5 \cdot \left(\color{blue}{im} \cdot \sqrt{\frac{1}{re}}\right) \]

      5. inv-pow 82.1%

         \[\leadsto 0.5 \cdot \left(im \cdot \sqrt{\color{blue}{{re}^{-1}}}\right) \]

      6. sqrt-pow1 82.1%

         \[\leadsto 0.5 \cdot \left(im \cdot \color{blue}{{re}^{\left(\frac{-1}{2}\right)}}\right) \]

      7. metadata-eval 82.1%

         \[\leadsto 0.5 \cdot \left(im \cdot {re}^{\color{blue}{-0.5}}\right) \]

   8. Applied egg-rr 82.1%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot {re}^{-0.5}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -6.4 \cdot 10^{+47}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{-9}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \]

Alternative 3: 76.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -6.4 \cdot 10^{+43}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 50000:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -6.4e+43)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 50000.0)
     (* 0.5 (sqrt (* 2.0 im)))
     (* 0.5 (* im (pow re -0.5))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -6.4e+43) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 50000.0) {
		tmp = 0.5 * sqrt((2.0 * im));
	} else {
		tmp = 0.5 * (im * pow(re, -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-6.4d+43)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 50000.0d0) then
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    else
        tmp = 0.5d0 * (im * (re ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -6.4e+43) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 50000.0) {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	} else {
		tmp = 0.5 * (im * Math.pow(re, -0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -6.4e+43:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 50000.0:
		tmp = 0.5 * math.sqrt((2.0 * im))
	else:
		tmp = 0.5 * (im * math.pow(re, -0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -6.4e+43)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 50000.0)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	else
		tmp = Float64(0.5 * Float64(im * (re ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -6.4e+43)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 50000.0)
		tmp = 0.5 * sqrt((2.0 * im));
	else
		tmp = 0.5 * (im * (re ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -6.4e+43], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 50000.0], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * N[Power[re, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -6.40000000000000029e43

   1. Initial program 23.4%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   2. Step-by-step derivation

      1. hypot-def 100.0%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\mathsf{hypot}\left(re, im\right)} - re\right)} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}} \]

   4. Taylor expanded in re around -inf 81.7%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-4 \cdot re}} \]
   5. Step-by-step derivation

      1. *-commutative 81.7%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   6. Simplified 81.7%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   if -6.40000000000000029e43 < re < 5e4

   1. Initial program 63.4%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   2. Step-by-step derivation

      1. hypot-def 90.6%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\mathsf{hypot}\left(re, im\right)} - re\right)} \]

   3. Simplified 90.6%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}} \]

   4. Taylor expanded in re around 0 74.1%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{2 \cdot im}} \]
   5. Step-by-step derivation

      1. *-commutative 74.1%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{im \cdot 2}} \]

   6. Simplified 74.1%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{im \cdot 2}} \]

   if 5e4 < re

   1. Initial program 10.4%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   2. Step-by-step derivation

      1. hypot-def 36.7%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\mathsf{hypot}\left(re, im\right)} - re\right)} \]

   3. Simplified 36.7%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}} \]

   4. Taylor expanded in re around inf 58.7%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{{im}^{2}}{re}}} \]
   5. Step-by-step derivation

      1. unpow2 58.7%

         \[\leadsto 0.5 \cdot \sqrt{\frac{\color{blue}{im \cdot im}}{re}} \]

   6. Simplified 58.7%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{im \cdot im}{re}}} \]
   7. Step-by-step derivation

      1. div-inv 58.7%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\left(im \cdot im\right) \cdot \frac{1}{re}}} \]

      2. sqrt-prod 71.5%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{im \cdot im} \cdot \sqrt{\frac{1}{re}}\right)} \]

      3. sqrt-prod 83.7%

         \[\leadsto 0.5 \cdot \left(\color{blue}{\left(\sqrt{im} \cdot \sqrt{im}\right)} \cdot \sqrt{\frac{1}{re}}\right) \]

      4. add-sqr-sqrt 83.9%

         \[\leadsto 0.5 \cdot \left(\color{blue}{im} \cdot \sqrt{\frac{1}{re}}\right) \]

      5. inv-pow 83.9%

         \[\leadsto 0.5 \cdot \left(im \cdot \sqrt{\color{blue}{{re}^{-1}}}\right) \]

      6. sqrt-pow1 84.0%

         \[\leadsto 0.5 \cdot \left(im \cdot \color{blue}{{re}^{\left(\frac{-1}{2}\right)}}\right) \]

      7. metadata-eval 84.0%

         \[\leadsto 0.5 \cdot \left(im \cdot {re}^{\color{blue}{-0.5}}\right) \]

   8. Applied egg-rr 84.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot {re}^{-0.5}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -6.4 \cdot 10^{+43}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 50000:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \]

Alternative 4: 76.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4.6 \cdot 10^{+45}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 0.0006:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -4.6e+45)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 0.0006) (* 0.5 (sqrt (* 2.0 im))) (* 0.5 (/ im (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -4.6e+45) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 0.0006) {
		tmp = 0.5 * sqrt((2.0 * im));
	} else {
		tmp = 0.5 * (im / sqrt(re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-4.6d+45)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 0.0006d0) then
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    else
        tmp = 0.5d0 * (im / sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -4.6e+45) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 0.0006) {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	} else {
		tmp = 0.5 * (im / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -4.6e+45:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 0.0006:
		tmp = 0.5 * math.sqrt((2.0 * im))
	else:
		tmp = 0.5 * (im / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -4.6e+45)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 0.0006)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	else
		tmp = Float64(0.5 * Float64(im / sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -4.6e+45)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 0.0006)
		tmp = 0.5 * sqrt((2.0 * im));
	else
		tmp = 0.5 * (im / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -4.6e+45], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 0.0006], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -4.60000000000000025e45

   1. Initial program 23.4%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   2. Step-by-step derivation

      1. hypot-def 100.0%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\mathsf{hypot}\left(re, im\right)} - re\right)} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}} \]

   4. Taylor expanded in re around -inf 81.7%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-4 \cdot re}} \]
   5. Step-by-step derivation

      1. *-commutative 81.7%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   6. Simplified 81.7%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   if -4.60000000000000025e45 < re < 5.99999999999999947e-4

   1. Initial program 64.3%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   2. Step-by-step derivation

      1. hypot-def 91.2%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\mathsf{hypot}\left(re, im\right)} - re\right)} \]

   3. Simplified 91.2%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}} \]

   4. Taylor expanded in re around 0 74.4%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{2 \cdot im}} \]
   5. Step-by-step derivation

      1. *-commutative 74.4%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{im \cdot 2}} \]

   6. Simplified 74.4%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{im \cdot 2}} \]

   if 5.99999999999999947e-4 < re

   1. Initial program 10.3%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   2. Step-by-step derivation

      1. hypot-def 37.2%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\mathsf{hypot}\left(re, im\right)} - re\right)} \]

   3. Simplified 37.2%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}} \]

   4. Taylor expanded in re around inf 57.1%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{{im}^{2}}{re}}} \]
   5. Step-by-step derivation

      1. unpow2 57.1%

         \[\leadsto 0.5 \cdot \sqrt{\frac{\color{blue}{im \cdot im}}{re}} \]

   6. Simplified 57.1%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{im \cdot im}{re}}} \]

   7. Taylor expanded in im around 0 83.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{\frac{1}{re}} \cdot im\right)} \]
   8. Step-by-step derivation

      1. unpow-1 83.0%

         \[\leadsto 0.5 \cdot \left(\sqrt{\color{blue}{{re}^{-1}}} \cdot im\right) \]

      2. metadata-eval 83.0%

         \[\leadsto 0.5 \cdot \left(\sqrt{{re}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \cdot im\right) \]

      3. pow-sqr 83.0%

         \[\leadsto 0.5 \cdot \left(\sqrt{\color{blue}{{re}^{-0.5} \cdot {re}^{-0.5}}} \cdot im\right) \]

      4. rem-sqrt-square 83.0%

         \[\leadsto 0.5 \cdot \left(\color{blue}{\left|{re}^{-0.5}\right|} \cdot im\right) \]

      5. rem-square-sqrt 82.7%

         \[\leadsto 0.5 \cdot \left(\left|\color{blue}{\sqrt{{re}^{-0.5}} \cdot \sqrt{{re}^{-0.5}}}\right| \cdot im\right) \]

      6. fabs-sqr 82.7%

         \[\leadsto 0.5 \cdot \left(\color{blue}{\left(\sqrt{{re}^{-0.5}} \cdot \sqrt{{re}^{-0.5}}\right)} \cdot im\right) \]

      7. rem-square-sqrt 83.0%

         \[\leadsto 0.5 \cdot \left(\color{blue}{{re}^{-0.5}} \cdot im\right) \]

      8. exp-to-pow 78.7%

         \[\leadsto 0.5 \cdot \left(\color{blue}{e^{\log re \cdot -0.5}} \cdot im\right) \]

      9. metadata-eval 78.7%

         \[\leadsto 0.5 \cdot \left(e^{\log re \cdot \color{blue}{\left(-0.5\right)}} \cdot im\right) \]

      10. distribute-rgt-neg-in 78.7%

          \[\leadsto 0.5 \cdot \left(e^{\color{blue}{-\log re \cdot 0.5}} \cdot im\right) \]

      11. exp-neg 78.7%

          \[\leadsto 0.5 \cdot \left(\color{blue}{\frac{1}{e^{\log re \cdot 0.5}}} \cdot im\right) \]

      12. exp-to-pow 82.9%

          \[\leadsto 0.5 \cdot \left(\frac{1}{\color{blue}{{re}^{0.5}}} \cdot im\right) \]

      13. unpow1/2 82.9%

          \[\leadsto 0.5 \cdot \left(\frac{1}{\color{blue}{\sqrt{re}}} \cdot im\right) \]

      14. associate-*l/ 83.0%

          \[\leadsto 0.5 \cdot \color{blue}{\frac{1 \cdot im}{\sqrt{re}}} \]

      15. *-lft-identity 83.0%

          \[\leadsto 0.5 \cdot \frac{\color{blue}{im}}{\sqrt{re}} \]

   9. Simplified 83.0%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{im}{\sqrt{re}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4.6 \cdot 10^{+45}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 0.0006:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]

Alternative 5: 63.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.35 \cdot 10^{+45}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.35e+45) (* 0.5 (sqrt (* re -4.0))) (* 0.5 (sqrt (* 2.0 im)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.35e+45) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else {
		tmp = 0.5 * sqrt((2.0 * im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-1.35d+45)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.35e+45) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.35e+45:
		tmp = 0.5 * math.sqrt((re * -4.0))
	else:
		tmp = 0.5 * math.sqrt((2.0 * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.35e+45)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.35e+45)
		tmp = 0.5 * sqrt((re * -4.0));
	else
		tmp = 0.5 * sqrt((2.0 * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.35e+45], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -1.34999999999999992e45

   1. Initial program 23.4%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   2. Step-by-step derivation

      1. hypot-def 100.0%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\mathsf{hypot}\left(re, im\right)} - re\right)} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}} \]

   4. Taylor expanded in re around -inf 81.7%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-4 \cdot re}} \]
   5. Step-by-step derivation

      1. *-commutative 81.7%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   6. Simplified 81.7%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   if -1.34999999999999992e45 < re

   1. Initial program 47.0%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   2. Step-by-step derivation

      1. hypot-def 73.9%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\mathsf{hypot}\left(re, im\right)} - re\right)} \]

   3. Simplified 73.9%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}} \]

   4. Taylor expanded in re around 0 57.8%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{2 \cdot im}} \]
   5. Step-by-step derivation

      1. *-commutative 57.8%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{im \cdot 2}} \]

   6. Simplified 57.8%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{im \cdot 2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.35 \cdot 10^{+45}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \]

Alternative 6: 51.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \sqrt{2 \cdot im} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* 0.5 (sqrt (* 2.0 im))))

C

double code(double re, double im) {
	return 0.5 * sqrt((2.0 * im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.5d0 * sqrt((2.0d0 * im))
end function

Java

public static double code(double re, double im) {
	return 0.5 * Math.sqrt((2.0 * im));
}

Python

def code(re, im):
	return 0.5 * math.sqrt((2.0 * im))

Julia

function code(re, im)
	return Float64(0.5 * sqrt(Float64(2.0 * im)))
end

MATLAB

function tmp = code(re, im)
	tmp = 0.5 * sqrt((2.0 * im));
end

Wolfram

code[re_, im_] := N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 42.4%

   \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
2. Step-by-step derivation

   1. hypot-def 79.0%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\mathsf{hypot}\left(re, im\right)} - re\right)} \]

3. Simplified 79.0%

   \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}} \]

4. Taylor expanded in re around 0 51.3%

   \[\leadsto 0.5 \cdot \sqrt{\color{blue}{2 \cdot im}} \]
5. Step-by-step derivation

   1. *-commutative 51.3%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{im \cdot 2}} \]

6. Simplified 51.3%

   \[\leadsto 0.5 \cdot \sqrt{\color{blue}{im \cdot 2}} \]

7. Final simplification 51.3%

   \[\leadsto 0.5 \cdot \sqrt{2 \cdot im} \]

Reproduce

herbie shell --seed 2023174 
(FPCore (re im)
  :name "math.sqrt on complex, imaginary part, im greater than 0 branch"
  :precision binary64
  :pre (> im 0.0)
  (* 0.5 (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re)))))